After the Battle of the Camps, what would have happened if Robb had kept the Riverlands army, called Roose Bolton down and besieged Tywin at Harrenhall? It’s not like Tywin could have held out long in Harrenhall given that he had 15,000 soldiers, plus staff, and not much in the way of supplies.

Stafford Lannister’s army would have marched down the River road and stormed through the Riverlands unopposed, leaving Robb cut off from his supply base and trapped between two enemy armies. Meanwhile, Robb would have had to somehow feed 40,000 men out in the field, which means his army is going to starve faster than Tywin’s will. 

Tywin wanted Robb to come after him at Harrenhal.

At the same time, keeping the Riverlords together would have been a good idea, by giving Robb another army to use in the field.